(expires 10/28)
Consider the differential equation
, where the vector
and the field
. Plot a few
solutions. What happens to them when ?
Give a ``Maple-proof'' that this is a general fact for every
solution. [A ``Maple-proof'' is an argument that is rigorous once we
accept Maple results as incontrovertibly true.]

18.

(expires 10/28) (No Maple.)
For the equation
,
, with the vector field

prove that the origin is an attractor in the future, i.e., every
solution verifies

[You can ask around how to do this, but then you have to show clearly
that you have understood it.]

19.

(expires 10/28)
For the system of differential equations of
prob. #23,

find the eigenvalues and eigenvectors of the Jacobian at the fixed points.
[This is a give-away if you have done #16.]

20.

(expires 10/28)
Consider the equations of the glider with no drag
term (). Use
dsolve, type=numeric to solve them numerically
with initial conditions , . Then solve exactly the
linearized system around the fixed point
, with
the same initial conditions. Graph the two functions for ,
and give a good estimate of their maximum difference. What happens if we
take a larger -range?